It isn’t valid to extend a log-log plot. A progression is valid by showing exponential growth along a linear time axis, so a graph with a linear x (time) axis and a log y axis can be validly extended (provided of course that one has analyzed the paradigm being measured and shown that it will not saturate to an asymptote)...In 2006 Kevin Kelly took up the issue and reposted Drum's extrapolation (click on the image to view a nontruncated version):
So the point of the log-log plot is simply to show that a phenomenon has in fact accelerated in the past. It is not valid to extend the line. For one thing the log-log plot cannot go into the future because that is the nature of the log time axis...
Kurzweil responded to Kelly like he had to Drum:You cite the extension made by Kevin Drum of the log-log plot that I provide of key paradigm shifts in biological and technological evolution...This extension is utterly invalid. You cannot extend in this way a log-log plot for just the reasons you cite. The only straight line that is valid to extend on a log plot is a straight line representing exponential growth when the time axis is on a linear scale and the a value (such as price-performance) is on a log scale...But it is not valid to extend the straight line when the time axis is on a log scale. The only point of these graphs is that there has been acceleration in paradigm shift in biological and technological evolution.Unfortunately, by the time Drum appended Kurzweil's email to his review, I had lost the URL. By chance I recently rediscovered it on Singularity disbeliever PZ Myers' blog.
What to make of all this? sigh I've already spent too much time--and not enough time--thinking about it so I'm going to stop arbitrarily. My apologies to Kurzweil and other authors & commenters if I'm unaware of relevant insights. The following is way too speculative for my liking, but here goes:
For the purposes of this post, I ignore Myers' contentions and accept Kurzweil's chart as valid. I assume that his plotted 'events' were chosen a priori without using a preconceived mathematical model. Upon scrutiny, the plot reveals interesting properties and implications[1].
a. It resembles a log-log plot of a power law. Curiously, TSIN's discussion of the figure does not mention the power-law interpretation.
b. There are constraints on the "times to the next event": between any two events A and B, the sum of the 'times to the next event' equals the total time elapsed between A and B. That is obvious of course, but it has implications for the spacing of events that obey a power law or any formula for that matter. (Cf. Myers' harsh criticism of Kurzweil's selection of events.)
c. The overall slope of the 'line' is more or less consistent with a value of -1. A(n approximately) unit slope is noteworthy (each event is half as close to the Singularity as its predecessor), but why not some other number? I won't give equations, but basic engineering math suffices for handwaving arguments that the number of events will multiply logarithmically as the Singularity is approached. A logarithmic Singularity is not Kurzweil's preferred scenario, but he mentions it on p. 495.
d. Like commenter dreish at KurzweilAI.net, I am unpersuaded by Kurzweil's criticism of Drum's extrapolation. Extending a line segment over a distance smaller than its length is plausible on its face. Kurzweil should make a better argument.
e. Drum's extrapolation begs a question that neither his post nor Kurzweil's response addresses. It seems reasonable to try to use 'countdown' data to forecast when the Singularity will occur. If that's not the case, why not? (Per 'dreish', consider the time axis, i.e. "years before today's date as of writing". If the 'date of writing' were the Singularity onset date, then events should have multiplied and clustered as the Singularity approached in real time; the x-axis of a log-time plot would have to be extended to display the real-time clustering. For a date of writing slightly before the Singularity onset date, I'd expect Kurzweil's existing log-log plot to depart from a straight line when the time between events becomes smaller than the time to the Singularity. [For a date of writing after the Singularity, the straight line would change to a "waterfall" near the onset time.] If the data is good enough, the deviation from a straight line might yield an estimate the time of the Singularity. Kurzweil does not discuss the issue. In fact, he describes the World Wide Web as an 'event' but, surprisingly, does not predict the time to the event that will follow the WWW.)
f. Afterthought. Note that if, as a function of the time to the Singularity, the time between events scales with a power less than one, the total number of events will necessarily be finite. (For sufficiently small positive x, x to a power is less than x if the power is less than one. Thus, it would be impossible to pack an infinite number of events close to each other as the Singularity approaches.) It's not clear how to interpret such a situation.
There is more to the Countdown plot than meets the eye at first glance. It's not entirely apparent whether the chart is meant to be descriptive or quantitative. If it's quantitative, it points to more information than has been extracted to date. Addressing Myers' criticism might be happen as a byproduct of digging such information out. How will the chart look when drawn at the Singularity? Will it retain its current linear form, or will the shape change? Either outcome would be interesting.
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[1] NB: in TSIN, the chart that Drum noticed is followed by over 600 pages including 100 pages of annotated footnotes. IMO the Drum-Kurzweil kerfluffle is interesting precisely because new things can be said without tangling with the interlinked sprawl of the overall book.
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